Title: A posteriori adaptive mesh technique with a priori error estimates for singularly perturbed semilinear parabolic convection-diffusion equations

Authors: G.I. Shishkin

Addresses: Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Ekaterinburg 620219, Russia

Abstract: Dirichlet problem is considered for a singularly perturbed semilinear parabolic convection-diffusion equation on a rectangular domain. The solution of the classical finite difference scheme on a uniform mesh converges at the rate O((ε + N⁻¹)⁻¹ N⁻¹ + N₀⁻¹) where N + 1 and N₀ + 1 denote the numbers of mesh points with respect to χ and t respectively, ε ∈ (0, 1] is the perturbation parameter. Using nonlinear and linearised basic classical schemes, finite difference schemes on a posteriori adaptive meshes based on uniform subgrids are constructed. The subdomains where grid refinement is required are defined by the gradients of the solutions of the intermediate discrete problems. The constructed difference schemes converge |almost ε-uniformly|, namely, at the rate O((ε^−υN⁻¹) + N^−1/2 + N₀⁻¹) where υ is an arbitrary number from (0, 1].

Keywords: singular perturbations; boundary value problems; semilinear parabolic convection-diffusion equations; finite difference approximation; a posteriori adaptive meshes; uniform subgrids; almost ε-uniform convergence; discrete solution gradient.

DOI: 10.1504/IJCSM.2007.016542

International Journal of Computing Science and Mathematics, 2007 Vol.1 No.2/3/4, pp.374 - 395

Published online: 07 Jan 2008 *

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